System and method for designing efficient super resolution deep convolutional neural networks by cascade network training, cascade network trimming, and dilated convolutions

ABSTRACT

Apparatuses and methods of manufacturing same, systems, and methods for generating a convolutional neural network (CNN) are described. In one aspect, a minimal CNN having, e.g., three or more layers is trained. Cascade training may be performed on the trained CNN to insert one or more intermediate layers until a training error is less than a threshold. When cascade training is complete, cascade network trimming of the CNN output from the cascade training may be performed to improve computational efficiency. To further reduce network parameters, convolutional filters may be replaced with dilated convolutional filters with the same receptive field, followed by additional training/fine-tuning.

PRIORITY

This is a Continuation of U.S. patent application Ser. No. 15/655,557,filed on Jul. 20, 2017 in the United States Patent and Trademark Office,which claims priority under 35 U.S.C. § 119(e) to U.S. ProvisionalPatent Application Ser. No. 62/471,816 filed on Mar. 15, 2017, theentire contents of which are incorporated herein by reference.

FIELD

The present disclosure relates generally to image super resolution, andmore particularly, to system and method for designing efficient superresolution deep convolutional neural networks by cascade networktraining, cascade network trimming, and dilated convolutions.

BACKGROUND

Super resolution imaging generates a high resolution (HR) image from alow resolution (LR) image. Super resolution (SR) has wide applicability,from surveillance and face/iris recognition to medical image processing,as well as the straightforward improvement of the resolution of imagesand video. Many algorithms/systems have been proposed for performing SR,from interpolations (Li, Xin and Orchard, Michael, New edge-directedinterpolation. IEEE Transactions on Image Processing (TIP), vol. 10,issue 10, pp. 1521-1527 (October 2001), which is incorporated byreference in its entirety), contour features (Tai, Yu-Wing; Liu,Shuaicheng; Brown, Michael; and Lin, Stephen, Super resolution usingedge prior and single image detail synthesis. 2010 IEEE Int'l Conferenceon Computer Vision and Pattern Recognition (CVPR), pp. 2400-2407, whichis incorporated by reference in its entirety), and statistical imagepriors (Kim, Kwang In and Kwon, Younghee. Single-image super-resolutionusing sparse regression and natural image prior. IEEE Transactions onPattern Analysis and Machine Intelligence (TPAMI), vol. 32, no. 6, pp.1127-1133 (January 2010), which is incorporated by reference in itsentirety), to example-based methods which learn from a dictionary ofpatches, such as neighbor embedding (Chang, Hong; Yeung, Dit-Yan; andXiong, Yimin, Super-resolution through neighbor embedding, 2004 CVPR,pp. 275-282, which is incorporated by reference in its entirety) andsparse coding (Yang, Jianchao; Wright, John; Huang, Thomas; and Ma, Yi,Image super-resolution via sparse representation. IEEE TIP, vol. 19, no.11, pp. 2861-2873 (November 2010), which is incorporated by reference inits entirety).

Recently, convolutional neural networks (CNNs) have provided asignificant improvement in SR accuracy. See, e.g., Dong, Chao; Loy, ChenChange; He, Kaiming; and Tang, Xiaoou, Learning a deep convolutionalnetwork for image super-resolution, 2014 European Conference on ComputerVision (ECCV), pp. 184-199 (hereinafter, “Dong et al. 2014”), which isincorporated by reference in its entirety. Sometimes referred to as“SRCNNs” (i.e., super-resolution convolutional neural networks), theiraccuracy can be limited by a small structure, e.g., 3-layers, and/orsmall context reception field. In response, researchers have proposedincreasing the size of SRCNNs, but most proposals use a prohibitivelylarge number of parameters, and many of the SRCNNs under discussioncannot be executed in real-time. Due to the large network sizes beingproposed, it can be very difficult to even guess at the appropriatetraining settings, i.e., learning rate, weight initialization, andweight decay. As a result, training may not converge at all or fall intoa local minimum.

SUMMARY

Accordingly, the present disclosure has been made to address at leastthe problems and/or disadvantages described herein and to provide atleast the advantages described below.

According to an aspect of the present disclosure, a method is providedwhich generates a convolutional neural network (CNN), including traininga CNN having three or more layers and performing cascade training on thetrained CNN to insert one or more intermediate layers into the CNN untila training error is less than a threshold, where the cascade training isan iterative process of one or more stages, in which each stageincludes: training the current CNN; determining whether the trainingerror is converging; and, if the training error is converging, insertinga preset number of intermediate layers in the CNN, the weights of eachnew layer being set to a predetermined setting; and starting a newstage.

According to an aspect of the present disclosure, a method is providedwhich generates a convolutional neural network (CNN), including traininga CNN having three or more layers and performing cascade networktrimming of the trained CNN, where the cascade network trimming is aniterative process of one or more stages, in which each stage includes:trimming a set number of layers of the current CNN by reducingdimensions of filters at one or more intermediate layers; determiningwhether the training error is converging; and, if the training error isconverging, determining whether all of the layers of the current CNNhave been trimmed; if all of the layers of the current CNN have beentrimmed, outputting the network trimmed CNN; and if all of the layers ofthe current CNN have not been trimmed, starting a new stage.

According to an aspect of the present disclosure, an apparatus isprovided for generating a convolutional neural network (CNN), includingone or more non-transitory computer-readable media and at least oneprocessor which, when executing instructions stored on one or morenon-transitory computer readable media, performs the steps of: traininga CNN having three or more layers; performing cascade training on thetrained CNN to add one or more intermediate layers until a trainingerror is less than a threshold; and performing cascade network trimmingof the CNN output from the cascade training.

According to an aspect of the present disclosure, a method is providedfor manufacturing a chipset which includes at least one processor which,when executing instructions stored on one or more non-transitorycomputer readable media, performs the steps of: training a CNN havingthree or more layers; performing cascade training on the trained CNN toadd one or more intermediate layers until a training error is less thana threshold; and performing network trimming of the CNN output from thecascade training; and the one or more non-transitory computer-readablemedia which store the instructions.

According to an aspect of the present disclosure, a method is providedfor testing an apparatus, including testing whether the apparatus has atleast one processor which, when executing instructions stored on one ormore non-transitory computer readable media, performs the steps of:training a CNN having three or more layers; performing cascade trainingon the trained CNN to add one or more intermediate layers until atraining error is less than a threshold; and performing cascade networktrimming of the CNN output from the cascade training; and testingwhether the apparatus has the one or more non-transitorycomputer-readable media which store the instructions.

BRIEF DESCRIPTION OF THE DRAWINGS

The above and other aspects, features, and advantages of certainembodiments of the present disclosure will be more apparent from thefollowing detailed description, taken in conjunction with theaccompanying drawings, in which:

FIG. 1 illustrates an exemplary block diagram of a method forconstructing a cascade trained super resolution convolutional neuralnetwork (CT-SRCNN), according to one embodiment;

FIG. 2 illustrates an exemplary diagram of cascade training, accordingto one embodiment;

FIGS. 3A and 3B illustrate some of the differences between existingtraining methods and cascade training according to an embodiment;

FIGS. 4A and 4B illustrate a beginning CNN and an ending CNN,respectively, after cascade training according to an embodiment of thepresent disclosure;

FIG. 5 illustrates an exemplary diagram of cascade network trimming,according to an embodiment of the present disclosure;

FIGS. 6A and 6B illustrate some of the differences between networktrimming methods, according to an embodiment of the present disclosure;

FIG. 7 illustrates an exemplary diagram for performing filter trimmingaccording to an embodiment of the present disclosure;

FIGS. 8A and 8B illustrate some of the differences between dilatedconvolution in accordance with an embodiment of the present disclosureand conventional convolution, respectively;

FIG. 9 illustrates an exemplary diagram of the present apparatus,according to one embodiment;

FIG. 10 illustrates an exemplary flowchart for manufacturing and testingthe present apparatus, according to one embodiment; and

FIG. 11 is an exemplary diagram illustrating the convergence speed ofcascade trained CNNs according to an embodiment of the presentdisclosure vs. non-cascade trained CNNs in experiments discussed in theAppendix hereto.

DETAILED DESCRIPTION

Hereinafter, embodiments of the present disclosure are described indetail with reference to the accompanying drawings. It should be notedthat the same elements are designated by the same reference numeralsalthough they are shown in different drawings. In the followingdescription, specific details such as detailed configurations andcomponents are merely provided to assist in the overall understanding ofthe embodiments of the present disclosure. Therefore, it should beapparent to those skilled in the art that various changes andmodifications of the embodiments described herein may be made withoutdeparting from the scope of the present disclosure. In addition,descriptions of well-known functions and constructions are omitted forclarity and conciseness. The terms described below are terms defined inconsideration of the functions in the present disclosure, and may bedifferent according to users, intentions of the users, or custom.Therefore, the definitions of the terms should be determined based onthe contents throughout the specification.

The present disclosure may have various modifications and variousembodiments, among which embodiments are described below in detail withreference to the accompanying drawings. However, it should be understoodthat the present disclosure is not limited to the embodiments, butincludes all modifications, equivalents, and alternatives within thescope of the present disclosure.

Although terms including an ordinal number such as first and second maybe used for describing various elements, the structural elements are notrestricted by the terms. The terms are only used to distinguish oneelement from another element. For example, without departing from thescope of the present disclosure, a first structural element may bereferred to as a second structural element. Similarly, the secondstructural element may also be referred to as the first structuralelement. As used herein, the term “and/or” includes any and allcombinations of one or more associated items.

The terms herein are merely used to describe various embodiments of thepresent disclosure but are not intended to limit the present disclosure.Singular forms are intended to include plural forms unless the contextclearly indicates otherwise. In the present disclosure, it should beunderstood that the terms “include” or “have” indicate existence of afeature, a number, a step, an operation, a structural element, parts, ora combination thereof, and do not exclude the existence or probabilityof addition of one or more other features, numerals, steps, operations,structural elements, parts, or combinations thereof.

Unless defined differently, all terms used herein have the same meaningsas those understood by a person skilled in the art to which the presentdisclosure belongs. Terms such as those defined in a generally useddictionary are to be interpreted to have the same meanings as thecontextual meanings in the relevant field of art, and are not to beinterpreted to have ideal or excessively formal meanings unless clearlydefined in the present disclosure.

Various embodiments may include one or more elements. An element mayinclude any structure arranged to perform certain operations. Althoughan embodiment may be described with a limited number of elements in acertain arrangement by way of example, the embodiment may include moreor less elements in alternate arrangements as desired for a givenimplementation. It is worthy to note that any reference to “oneembodiment” or “an embodiment” means that a particular feature,structure, or characteristic described in connection with the embodimentis included in at least one embodiment. The appearance of the phrase“one embodiment” (or “an embodiment”) in various places in thisspecification does not necessarily refer to the same embodiment.

This disclosure provides a new approach, or more accurately, several newtechniques, for creating an SRCNN. Herein, the term “cascade trainedsuper resolution convolutional neural network” (CT-SRCNN) may refer toall of the new techniques described herein together, or to one or moreof the new techniques, which should be made clear by the context inwhich the term is used. Different from existing approaches that trainall the layers from the beginning with unsupervised weightinitialization, CT-SRCNN starts training with a small network (e.g., 3layers). New layers are gradually inserted into the network when thecurrent network cannot adequately reduce the training error.

With this “cascade training” strategy, convergence is made easier, andthe accuracy is consistently increased when more layers are used. Butwhile the depth increases, the relative complexity of the network doesnot, because of the nature of the new layers. More specifically, all theweights of the new layers in CT-SRCNN are randomly initialized, and thelearning rate is fixed. This is a great advantage compared to approacheswhich need to spend a great deal of time and resources tuning theparameters. One specific example of a CT-SRCNN with 13 layers (as shownand discussed further below), the accuracy is competitive with thestate-of-the-art image SR networks, while having an execution speed morethan 5 times faster, and using only ⅕^(th) of the parameters.

In this disclosure, “cascade network trimming” is described, whichfurther refines the CT-SRCNN model by reducing the storage andcomputational complexities, as well as another method to further improvethe efficiency of super resolution deep convolutional neural networks bydeploying a form of “dilated convolution,” instead of performing thecomplete conventional convolutional calculations, which may furtherreduce the CT-SRCNN model complexity.

The rest of the disclosure discusses these three differentschemes/features of the CT-SRCNN in order:

I. Cascade training;

II. Cascade network trimming; and

III. Dilated convolution.

Although these three methods/techniques are discussed in the context ofthe CT-SRCNN, each method/technique could be applied individually orseverally to other SR schemes or CNN networks, as would be understood byone of ordinary skill in the art.

FIG. 1 illustrates an exemplary block diagram of a method forconstructing a cascade trained super resolution convolutional neuralnetwork (CT-SRCNN), according to one embodiment.

At 110, the training set is prepared, meaning a set of low resolution(LR) images with corresponding high resolution (HR) images by which theCT-SRCNN “learns” a model to use when attempting to create highresolution images from low resolution images. In this embodiment, at120, each LR image is bicubic upsampled and the LR/HR patches cropped inpreparation for training. See, e.g., Dong et al. 2014 and Dong, Chao;Loy, Chen Change; He, Kaiming; and Tang, Xiaoou, Image super-resolutionusing deep convolutional networks, IEEE Transactions on Pattern Analysisand Machine Intelligence (TPAMI), vol. 38, no. 2, pp. 295-307 (February2016) (hereinafter, “Dong et al. 2016a”), which is incorporated byreference in its entirety, for more details concerning this step. Aswould be understood by one of ordinary skill in the art, there are avariety of pre-training preparation techniques and this disclosure isnot limited to this bicubic upsampling and LR/HR patching as apre-training preparation technique.

At 130, cascade training is performed in accordance with the presentdisclosure. Embodiments of cascade training according to specificembodiments of the present disclosure are described below. At 140,cascade network trimming is performed in accordance with the presentdisclosure. Embodiments of network trimming according to specificembodiments of the present disclosure are further described below. At150, the process is complete and the CT-SRCNN system is ready for realworld use.

Although these different processes (i.e., cascade training and cascadenetwork trimming) are described and shown in FIG. 1 as separate anddistinct stages/steps, there may be overlap between these functions inactual implementations in accordance with the present disclosure.

I. Cascade Training

FIG. 2 illustrates an exemplary diagram of cascade training, accordingto one embodiment of the present disclosure. At 205, the process oftraining begins.

At 210, the training starts at stage i=1. The fledgling network startswith b number of layers, and C number of layers are added in each stagewhere the training error converges (220) or remains higher than athreshold (250). Thus, at each training stage i, a CNN with c*(i−1)+blayers is trained. When stage i=1, the CNN with the first b number oflayers is trained. After stage i=1, cascade training starts to addintermediate layers to the b number of layers, specifically C number oflayers at a time, as necessary.

At 220, it is determined whether the network has started to converge,e.g., whether the training error has stopped decreasing by a certainamount (from the previous stage). If it has (i.e., the CNN isconverging), C number of intermediate layers are added at 230, and thenext iteration begins at 240 (i=i+1). During this iterative process, thenew layers may be set to any arbitrary weighting, as the intermediatelayers will have no influence on the weight matrix sizes of the otherlayers. Indeed, all existing layers inherit their previous weightmatrix. This cascade training iterative process continues, making theCNN deeper and deeper, until the training error is smaller than athreshold at 250, and then the CNN model is output at 255.

FIGS. 3A and 3B illustrate some of the differences between cascadetraining and existing training methods.

In FIG. 3A, an example of the flowchart in FIG. 2 is shown. In FIG. 3A,b number of layers equals three, as shown at the top (310), whichrepresents the first CNN to be trained, and the number of layers C addedin each stage is one. Each new layer has its weights set randomly, whileeach pre-existing layer inherits its weights from the previous stage.With every newly inserted intermediate layer, the CNN becomes deeper. Ateach stage, the deeper CNN is trained again. Since most of the weightsare inherited from the previous stage, the continuous re-training isrelatively easy, even with a fixed learning rate.

Existing methods, however, as shown in FIG. 3B, start with a “complete”set of layers which need to be tuned at the same time. Training all ofthe layers at the same time as shown in FIG. 3B is far more complex thanthe scheme shown in FIG. 3A due to the slow convergence, where cascadetraining trains shallower networks until convergence, incrementallyinserts layers with random weights while keeping previously trainedlayers intact, and retrains the whole network until a deeper networkconverges. Moreover, cascade training can simply fix the learning rateand generate new layers with random weights.

FIGS. 4A and 4B illustrate a beginning CNN and an ending CNN,respectively, after cascade training according to an embodiment of thepresent disclosure.

Let x denote an interpolated LR image and y denote its matching HRimage. Given a training set {(x_(i),y_(i)), i=1, . . . , N} with Nsamples, the goal for the CT-SRCNN is to learn a model g that predictsthe HR output ŷ=g(x). During training, mean square error (MSE) ½Σ_(i=1)^(N)∥y_(i)−ŷ_(i)∥ is minimized over the training set.

In FIG. 4A, cascade training starts from a 3-layer model (b=3). Thefirst layer consists of 64 9×9 filters (410), and the second (413) andthe third layer (415) consist of 32 5×5 filters. All the weights (of newlayers) are randomly initialized by a Gaussian function with σ=0.001,and all convolutions have stride one. “Stride” is one of thehyperparameters of a convolutional layer, and controls how the depthcolumns around the spatial dimensions (width and height) areallocated—to put it another way, stride indicates how the filterconvolves around the input volume, namely, “stride one” indicates thatthe filter convolves around the input volume one pixel at a time,“stride two” indicates the filter convolves two pixels at time, etc.See, e.g., Definition of “Convolutional neural network,” downloaded onJun. 5, 2017 from Wikipedia athttps://en.wikipedia.org/wiki/Convolutional_neural_network; “ABeginner's Guide to Understanding Convolutional Networks—Part 2,”downloaded on Jun. 5, 2017 fromhttps://adeshpande3.github.io/A-Beginner%27s-Guide-To-Understanding-Convolutional-Neural-Networks-Part-2/;both of which are incorporated by reference in their entireties.

Returning to FIG. 4A, when the MSE of the current stage stops decreasingsignificantly, e.g., the error decreases less than 3% in an epoch, thetraining goes to the next stage. See, e.g. step 220 of FIG. 2. Toaccelerate the training in this embodiment, two new layers are insertedinto the network for each stage (i.e., c=2 in step 230 in FIG. 2). Thus,the training starts from 3 layers, as shown at FIG. 4A, and thenproceeds to 5 layers, 7 layers, . . . , and finally 13 layers after five(5) stages. Each new layer consists of 32 3×3 filters. This size ensuresa smaller network even when the CNN is becoming progressively deeper.The new intermediate layers are inserted immediately before the last 325×5 filters layer 415. The weights from any layer existing in thepreceding stage inherits the weights from the previous stage, and theweights of the two new layers are always randomly initialized (Gaussiandistribution with σ=0.001). Since new convolutional layers will reducethe size of the feature map, 2 pixels are zero-padded in each newintermediate 3×3 layer. As a result, all the stages in cascade traininghave the same size of the output, so that the training samples can beshared.

As a network goes deeper, it usually becomes more difficult for thetraining with existing methods to converge. For example, the SRCNN inDong et al. 2016a failed to show superior performance with more thanthree layers. In Kim, Jiwon; Lee, Jung Kwon; and Lee, Kyoung Mu,Accurate image super-resolution using very deep convolutional networks,2016 CVPR, pp. 1646-1654, which is incorporated by reference in itsentirety (hereinafter, “VDSR”), a high initial learning rate is tunedand gradually decreased. But when using a large diverse training set(e.g., more than 30 million patches from 160,000 images), the highlearning rate does not work well. A potential reason for this is thatthe high learning rate leads to vanishing/exploding gradients.

In CT-SRCNN, only a few weights are randomly initialized in each stage,so the convergence is relatively easy. A fixed learning rate 0.0001 forall layers in CT-SRCNN without any decay is also feasible. In order toaccelerate the training, only the first stage need be changed, e.g., thelearning rate of the first stage can be set to 0.001. Inexperiments/simulations, the 13-layer CT-SRCNN like the one in FIG. 4Bhas already achieved state-of-the-art accuracy, while using many lessparameters compared to other networks such as VDSR or Kim, Jiwon; Lee,Jung Kwon; and Lee, Kyoung Mu, Deeply-recursive convolutional networkfor image super-resolution, 2016 CVPR, pp. 1637-1645, which isincorporated by reference in its entirety (hereinafter, “DRCN”). To thecontrary, direct training of a randomly initialized deeper networkrequires a lot of effort in parameter tuning to ensure best convergencein these other networks, even though experiments have shown thesenetworks may fail to converge with acceptable error.

As shown in Table 1 below, when two image qualities metrics, the peaksto noise ratio (PSNR) and the structure similarity measure (SSIM), aremeasured, it can be seen that the CT-SRCNN achieves better quality andfaster speed. Moreover, the CT-SRCNN retrieves more details compared toVDSR and DRCN.

Given an L-layer CNN, assume the i^(th) layer has n_(i−1) inputchannels, a k_(i)×k_(i) convolution kernel, and n_(i) filters. Thenumber of parameters in the i^(th) layer is n_(i−1)×n_(i)×k_(i)×k_(i).The bias term is ignored in this calculation. Then the overall number ofparameters is Σ_(i=1) ^(L)n_(i−1)×n_(i)×k_(i)×k_(i). Thus, for example,in a 3-layer CT-SRCNN with 64-32-1 filters in each layer, n₀=1, n₁=64,n₂=32, n₃=1, k₁=9, k₂=5, k₃=5, so the overall number of parameters is1×64×9×9+64×5×5×32+1×32×5×5×1=57,184.

PSNR/SSIM are utilized to measure the image reconstruction quality. PSNRis the ratio between the maximum possible power of an image pixel andthe power of corrupting noise that affects the fidelity. It iscalculated as

${{P\; S\; N\; R} = {20\log_{10}\frac{255}{\sqrt{MSE}}}},$

where the MSE is calculated between the ground truth and a reconstructedimage (SR output). The larger the PSNR, the better the image quality.The maximum value of PSNR is infinite. See, e.g., definition of “Peaksignal-to-noise ratio,” downloaded on Jun. 27, 2017 from Wikipedia athttps://en.wikipedia.org/wiki/Peak_signal-to-noise_ratio, which isincorporated by reference in its entirety.

SSIM is a perception-based model that considers image degradation asperceived change in structural information, while also incorporating theluminance masking and contrast masking. It is more consistent with humanvision than PSNR. SSIM is calculated as

${{S\; S\; I\; M} = {20\log_{10}\frac{\left( {{2\mu_{x}\mu_{y}} + c_{1}} \right)\left( {{2\sigma_{xy}} + c_{2}} \right)}{\left( {\mu_{x}^{2} + \mu_{y}^{2} + c_{1}} \right)\left( {\sigma_{x}^{2} + \sigma_{y}^{2} + c_{2}} \right)}}},$

where x is the reconstructed image, y is the reference image (groundtruth), μ is the mean, σ is the variance, σ_(xy) is the covariancebetween x and y, c₁=6.5025, and c₂=58.5225. SSIM lays between [0,1]. Ifx is a perfect copy of y, the SSIM will be 1. See, e.g., definition of“Structural Similarity,” downloaded on Jun. 27, 2017 from Wikipedia athttps://en.wikipedi.org/wiki/Structural_similarity, which isincorporated by reference in its entirety.

TABLE I Comparison of CT-SRCNN and existing approaches Number of Timeper image Parameters PSNR SSIM (in seconds) VDSR   >600,000 29.77 0.83140.17 DRCN >1,000,000 29.76 0.8311 4.19 13-layer Cascade Trained  ~150,000 29.91 0.8324 0.03 (only) SRCNN Cascade trimmed 13-layer  ~120,000 29.91 0.8322 0.02 CT-SRCNN

II. Cascade Network Trimming

Most neural networks have redundancy. Removing such redundancy clearlyimproves efficiency. In embodiments of the present disclosure, a majornumber of filters and/or weights may be removed from certain layers witha minor loss in accuracy.

This technique/approach (cascade network trimming) can be used with thecascade training described above, or can be used independent of cascadetraining. Given a deep convolutional neural network with acceptableaccuracy or performance, techniques/approaches for reducing networksize, computational complexity, and/or processing speed while keepingthe network depth the same and not degrading the accuracy are alwaysneeded.

Similar to cascade training, cascade network trimming also includes aniterative process. In each stage, filters are trimmed from only dlayers, which means that, for an L-layer network, the (L−(i−1)d−1)thlayer to (L-id)th layer are trimmed in stage i. For example, whentrimming d=2 layers from a 13-layer CT-SRCNN, the 12^(th) and 11^(th)layers are trimmed in the first stage i=1, and then the network isfine-tuned. When it converges, the second stage i=2 begins with trimmingthe 9^(th) and 10^(th) layers. This procedure is iteratively repeateduntil all of the layers are trimmed. Although the 13^(th) layer isignored in the above procedure, the procedure may also be considered astrimming 12^(th) and 13^(th) layer in the first stage, and trimming10^(th) and 11^(th) layer in the second stage, etc.

FIG. 5 illustrates an exemplary diagram of cascade network trimming,according to one embodiment of the present disclosure. At 505, theprocess of trimming begins with a trained CNN with L layers.

At 510, the trimming starts at stage i=1. As mentioned above, only dlayers of the total L-layer CNN are trimmed in a stage. Thus, the(L−(i−1)d−1)th layer to (L-id)th layer are trimmed in stage i at 510. At520, fine tuning is performed. At 530, it is determined whether thetraining error has stopped decreasing by a certain amount (from theprevious stage). If it has, it is determined whether the total number ofstages multiplied by the layers trimmed per stage is greater than orequal to the total number of layers at 540 (“(id>=L)?”). If the trainingerror has not stopped decreasing at 530, the method returns to finetuning at 520.

If it is determined that the total number of stages multiplied by thelayers trimmed per stage is greater than or equal to the total number oflayers at 540 (“(id>=L)?”), the process ends and the trimmed CNN modelis output at 565. If it is determined that the total number of stagesmultiplied by the layers trimmed per stage is less than the total numberof layers at 540 (“(id>=L)?”), the method begins the next stage at 550(“i=i+1”).

FIGS. 6A and 6B illustrate some of the differences between networktrimming methods in accordance with an embodiment of the presentdisclosure.

In FIG. 6A, one layer of the CNN is trimmed per stage, and fine tuningis performed between each stage, in accordance with an embodiment of thepresent disclosure. By contrast, all of the layers of the CNN in FIG. 6Bare both fine-tuned and trimmed at the same time. Tuning and trimmingall of the layers at the same time as shown in FIG. 6B is far morecomplex than the scheme shown in FIG. 6A.

Cascade network trimming is done by trimming whole filters from thelayers. To recover any lost accuracy, trimming is done layer by layer,with fine-tuning till convergence after each trimmed layer or group oflayers.

As shown in FIG. 7, once a filter is trimmed, the adjacent layer willalso be influenced. In FIG. 7, a filter 710 (block of dotted lines) istrimmed from the ith layer, n_(i)=n_(i)−1, some weights 720 (indicatedby the dotted lines within the filters) in the (i+1)th layer will alsobe trimmed. So trimming the filter in the ith layer will reduce thecomputational cost for both the ith and the (i+1)th layer. In CNN, thenumber of input channels of the (i+1)th layer is equal to the number offilters (output channel) of the ith layer.

In FIG. 7, assume there are n_(i)=4 filters and n_(i−1)=5 input channelsin the ith layer, and n_(i+1)=10 filters and n_(i)=4 input channels inthe (i+1)th layer before the trimming. If filter 710 is trimmed from theith layer, the trimmed n_(i) will be reduced to 3, and the n_(i+1) isstill 10. The slices 720 in the (i+1)th layer are the trimmed weights,which correspond to the multiplications. As mentioned in the lastsection, there will be n_(i−1)×k_(i)×k_(i)×n_(i)×w_(i)×h_(i)multiplications in the ith layer, andn_(i)×k_(i+1)×k₁₊₁×n_(i+1)×w_(i+1)×h_(i+1) multiplications in the(i+1)th layer. Since n_(i) is reduced, the number of multiplications inboth the ith layer and (i+1)th layer are also reduced.

An appropriate criteria is used to decide which filters are to betrimmed. In this embodiment, a measurement of relative importance isused. More specifically, the relative importance R_(i,j) of the jthfilter in the ith layer is defined by the square sum of all the weightsin jth filter, where W_(i,j) is the weights matrix of the jth filter inthe ith layer, as shown in Equation (1):

$\begin{matrix}{R_{i,j} = {\sum\limits_{w \in W_{i,j}}w^{2}}} & (1)\end{matrix}$

Accordingly, the filters with the smallest R_(i,j) are removed. Asdiscussed above, when filter 710 is trimmed from the ith layer, someweights 720 in the (i+1)th layer will also be trimmed, resulting inW′_(i+1,j). Thus, when calculating R_(i+1,j), either the non-trimmedweights W_(i+1,j) (also referred to as “independent trimming”) are usedin Equation (3), or the trimmed weights W′_(i+1,j) are used in Equation(2):

$\begin{matrix}{R_{{i + 1},j} = {\sum\limits_{w \in W_{{i + 1},j}^{\prime}}w^{2}}} & (2)\end{matrix}$

The algorithm below provides an exemplary high-level description of theiterative process for trimming filters from the layers.

Algorithm for Trimming Filters Parameters ∈_(filters,i), i = 1, . . . ,L rate of filter trimming for each layer Input: CT-SRCNN model with Llayers, each layer has M_(i) filters 1. Repeat for i = 1, 2, . . . , L 1.1 Calculate R_(i,j), j = 1, . . . , M_(i) for all the filters in the ith layer using (3) or (4)  1.2 Remove the ∈_(filters,i) × M_(i)filters from the ith layer  1.3 If i < L, remove the correspondingweights in  i + 1th layer 2. Fine-tuning and output trimmed model

With different rates/thresholds ∈_(weights) and ∈_(filters,i), differenttrimmed models may be created. Since filter trimming influences theadjacent layers, fine-tuning will be needed to retrieve the accuracy inmost cases where filter trimming is used. By contrast, weight pruninghas a relatively smaller influence. With an appropriate trimming rate(e.g., less than 0.2), the accuracy will not decrease much even withoutfine-tuning.

III. Dilated Convolution

Dilated convolution, also known as à trous convolution, is a type ofconvolution which was originally developed for wavelet decomposition(see Holschneider, M.; Kronland-Martinet, R.; Morlet, J.; andTchamitchian, Ph., A Real-Time Algorithm for Signal Analysis with theHelp of the Wavelet Transform in WAVELETS: TIME-FREQUENCY METHODS ANDPHASE SPACE, J. M. Combes et al., eds., pp. 286-297 (1987), which isincorporated by reference in its entirety), but has been applied tosemantic segmentation, particularly in order to get dense features (see,e.g., Yu, Fisher and Koltun, Vladlen, Multi-scale context aggregation bydilated convolutions, 2016 Int'l Conference on Learning Representations(ICLR) (hereinafter, “Yu et al. 2016”), which is incorporated byreference in its entirety).

In a purely convolutional network composed of convolutions layerswithout pooling, the receptive field of units can only grow linearlylayer by layer because the feature maps are generated based onconvolving adjacent pixels from the input. A feasible way to increasethe receptive field is to convolve the input pixels from a largerregion. This is similar to using a ‘dilation kernel’ in dilationconvolution instead of using the conventional dense kernel forconventional convolution.

Suppose F is a discrete function, K is a convolution kernel, and thedilated convolution *_(d) is a generalized version of typicalconvolution, as defined by Equation (3) below, where d is the dilationfactor. The conventional convolution is a simple 1-dilated convolution(i.e., when d=1).

$\begin{matrix}{{F_{*d}{K(z)}} = {\sum\limits_{{x + {dy}} = z}{{F(x)}{K(y)}}}} & (3)\end{matrix}$

One advantage of applying dilated convolution in a CNN is that thedilated version has a larger reception field, as illustrated in FIGS. 8Aand 8B. The dilated convolutional filter is obtained by upsampling theoriginal filter, i.e., by inserting zeros between its elements. Hence,by design the dilated filter has a structured pattern of zero elements.Compared to weight pruning, where the zero elements have a randompattern and location, dilated filters have a structured pattern for thezero weights, and are much more useful for reducing the computationalcomplexity in hardware and software. Hence, for super resolution,embodiments of the present disclosure deploy the dilated filtersdifferently than their typical usage, by keeping the same receptivefield and instead using it to reduce the computational complexity incomparison to the non-dilated filter with the same receptive field.

FIGS. 8A and 8B illustrate some of the differences between dilatedconvolution in accordance with an embodiment of the present disclosureand conventional convolution, respectively. In FIG. 8B, conventionalconvolution is performed with stride two, while, in FIG. 8A, there is a2-dilated convolution (meaning that the multiplication-and-accumulationoperation in the convolution is applied every 2 pixels, instead of everypixel) with stride one according to an embodiment of the presentdisclosure. Although FIGS. 8A and 8B have the same feature map size(with padding for the dilated version), the reception field of the2-dilated feature map is larger compared to the convolutional one. In aCNN, the input and output are 2-D feature maps so FIG. 8A or 8B are onthe x-direction or y-direction only.

FIG. 8B illustrates an example of a conventional convolution with a size3 kernel and stride 2, where the input is a 7-pixel signal (representedby 7 circles). In FIG. 8B, every 3 adjacent pixels are convolved (asindicated by the connecting lines) with the kernel and then generate anoutput (a square) of the feature map, beginning with the 1^(st) to3^(rd) pixels (the lined circles) and the first output (the linedsquare) of the feature map The next convolution in FIG. 8B are the3^(rd) to 5^(th) pixels because the stride is 2, and the next output(the black square) of the feature map consists of 3 elements, withreceptive field 3.

By contrast, FIG. 8A illustrates an example of a 2-dilated convolutionwith a size 3 kernel and stride 1. In d-dilated convolution, theconvolution is applied every d pixels. So the first output (the linedsquare) of the feature map is generated by convolving the 1^(st),3^(rd), and 5^(th) pixels (lined circles) with the 3×3 kernel. Then thenext output (the black square) is generated by convolving the 2^(nd),4^(th), and 6^(th) pixels.

In an embodiment where all the layers in the CNN are convolutions withstride one, the dilated convolution may be applied in a different way.Given a k×k convolution kernel with stride one, the reception field ofthe resulting feature map is k×k. If 2-dilated convolution is used, thereception field of the resulting feature map is (2k−1)×(2k−1). Forexample, the 9×9 1-dilated layer 410 and a 5×5 1-dilated layer 413 ofthe CT-SRCNN in FIGS. 4A and 4B may be replaced by a 5×5 2-dilated layerand 3×3 2-dilated layer, respectively, instead. The resulting networkwill have the same size reception field, but less parameters due to thesmaller kernel size.

Accordingly, in one embodiment, once a CT-SRCNN with a 9×9 1-dilatedlayer and two 5×5 1-dilated layers is trained, those layers may bereplaced by a 5×5 2-dilated layer and two 3×3 2-dilated layers beforefine tuning is performed. Unlike Yu et al. 2016, a dilated CT-SRCNNaccording to an embodiment of the present disclosure does not need anyzero padding in the dilated layer.

As mentioned above, many researchers are attempting to increase theaccuracy and efficiency of SRNNs by, for example, using more layers(e.g., VDSR) or a deeply recursive structure (e.g., DRCN). Otherresearchers similarly propose to use more complicated networks. Wang,Zhaowen; Liu, Ding; Yang, Jianchao; Han, Wei; and Huang, Thomas, Deepnetworks for image super-resolution with sparse prior, 2015 IEEE Int'lConference on Computer Vision (ICCV), pp. 370-378, which is incorporatedherein by reference, integrated a sparse representation prior withfeed-forward network based on the learned iterative shrinkage andthresholding algorithm. VDSR increased the number of layers to 20 andused small filters and a high learning rate with adjustable gradientclipping; the same group also designed a deep recursive CNN withrecursive-supervision and skip-connection in DRCN. Dahl, Ryan; Norouzi,Mohammad; and Shlens, Jonathon, Pixel Recursive Super Resolution, arXiv1702.00783 [22 Mar. 2017], which is incorporated herein by reference,combined the ResNet with a Pixel Recursive Super Resolution, whichshowed promising results on face and bed SR where super resolution isapplied to bed images).

Others prefer to use perception loss instead of the mean square error(MSE) for the training error, which is closer to natural texture andhuman vision. Sønderby, Casper; Caballero, Jose; Theis, Lucas; Shi,Wenzhe; and Huszár, Ferenc, Amortised MAP Inference for ImageSuper-resolution, arXiv 1610.04490 [21 Feb. 2017], which is incorporatedherein by reference, introduced a method for amortised MAP inference,which calculated the MAP estimation directly using CNN. Johnson, Justin;Alahi, Alexandre; and Fei-Fei, Li, Perceptual losses for real-time styletransfer and super-resolution, 2016 ECCV, pp. 694-711, which isincorporated herein by reference, proposed the use of perceptual lossfunctions for training feedforward networks for image transformationtasks. Ledig, Christian, et al., Photo-realistic single imagesuper-resolution using a generative adversarial network, arXiv1609.04802 [13 Apr. 2017], which is incorporated herein by reference,employed a very deep residual network (ResNet), and further presentedthe super resolution generative adversarial network (SRGAN) to obtainthe images similar to natural texture.

However, although the works listed above improved the accuracy of the SRsystem, the improved accuracy was at the cost of having morelayers/parameters and/or more difficult hyperparameter tuningprocedures. In other words, any advance in accuracy was counter-balancedby extreme increases in complexity.

Other researchers focused on improving efficiency by extracting thefeature maps in LR space and training using upscaling filters. Shi,Wenzhe, et al., Real-time Single Image and Video Super-Resolution Usingan Efficient sub-pixel convolutional neural network, 2016 CVPR, pp.1874-1883, which is incorporated herein by reference, introduced anefficient subpixel convolutional layer which learned an array ofupscaling filters to upscale the LR feature maps into the HR output.Dong, Chao; Loy, Chen Change; and Tang, Xiaoou, Accelerating thesuper-resolution convolutional neural network. 2016 ECCV, pp. 391-407,which is incorporated herein in its entirety (hereinafter, “Dong et al.2016b”), re-designed the SRCNN by adding smaller filters, adeconvolution layer, and feature space shrinkage to accelerate the speedwithout losing the accuracy.

However, due to the use of the upscaling layer, the patch size andcontext reception field of these networks will be relatively small. As aresult, the accuracy is relatively lower compared to extracting featuremaps from upsampled LR space.

By contrast, the CT-SRCNN described herein can go deeper, therebyachieving high accuracy, without the heavy tuning of parameters. Thenetwork size of the CT-SRCNN is much smaller compared to thestate-of-the-art solutions, such as those listed above. The CT-SRCNN canalso process 20-25 frames/second of video with a resolution of 720×480in a single GPU. This efficiency could be further enhanced by networktrimming and dilated convolution.

In this disclosure, a cascade training method is described which trainsdeep CNN for super resolution with both high accuracy and efficiency.The cascade training ensures that the network might consistently godeeper with a relatively smaller size. The network trimming and dilatedconvolution described herein further reduce the network complexity. Theexperimental results on benchmark image and video datasets show that thedisclosed method herein achieves competitive performance compared toother state-of-the-art solutions, but at much higher speeds.

Although described in the framework of image super-resolution, thetechniques described herein can be generalized any type of CNN for anytype of purpose, such as denoising, or image restoration.

FIG. 9 illustrates an exemplary diagram of the present apparatus,according to one embodiment. An apparatus 900 includes at least oneprocessor 910 and one or more non-transitory computer readable media920. The at least one processor 910, when executing instructions storedon the one or more non-transitory computer readable media 920, performsthe steps of training a CNN having three or more layers; performingcascade training on the trained CNN to add one or more intermediatelayers until a training error is less than a threshold; and performingnetwork trimming of the CNN output from the cascade training. Moreover,the one or more non-transitory computer-readable media 1020 storesinstructions for the at least one processor 910 to perform the steps oftraining a CNN having three or more layers; performing cascade trainingon the trained CNN to add one or more intermediate layers until atraining error is less than a threshold; and performing network trimmingof the CNN output from the cascade training.

FIG. 10 illustrates an exemplary flowchart for manufacturing and testingthe present apparatus, according to one embodiment.

At 1050, the apparatus (in this instance, the chipset described above)is manufactured, including at least one processor and one or morenon-transitory computer-readable media. When executing instructionsstored on the one or more non-transitory computer readable media, the atleast one processor performs the steps of training a CNN having three ormore layers; performing cascade training on the trained CNN to add oneor more intermediate layers until a training error is less than athreshold; and performing network trimming of the CNN output from thecascade training. The one or more non-transitory computer-readable mediastore instructions for the at least one processor to perform the stepsof training a CNN having three or more layers; performing cascadetraining on the trained CNN to add one or more intermediate layers untila training error is less than a threshold; and performing networktrimming of the CNN output from the cascade training.

At 1060, the apparatus (in this instance, a chipset) is tested. Testing1060 includes testing whether the apparatus has at least one processorwhich, when executing instructions stored on one or more non-transitorycomputer readable media, performs the steps of training a CNN havingthree or more layers; performing cascade training on the trained CNN toadd one or more intermediate layers until a training error is less thana threshold; and performing network trimming of the CNN output from thecascade training; and testing whether the apparatus has the one or morenon-transitory computer-readable media which store instructions for theat least one processor to perform the steps of training a CNN havingthree or more layers; performing cascade training on the trained CNN toadd one or more intermediate layers until a training error is less thana threshold; and performing network trimming of the CNN output from thecascade training.

The steps and/or operations described above in relation to an embodimentof the present disclosure may occur in a different order, or inparallel, or concurrently for different epochs, etc., depending on thespecific embodiment and/or implementation, as would be understood by oneof ordinary skill in the art. Different embodiments may perform actionsin a different order or by different ways or means. As would beunderstood by one of ordinary skill in the art, some drawings aresimplified representations of the actions performed, their descriptionsherein simplified overviews, and real-world implementations would bemuch more complex, require more stages and/or components, and would alsovary depending on the requirements of the particular implementation.Being simplified representations, these drawings do not show otherrequired steps as these may be known and understood by one of ordinaryskill in the art and may not be pertinent and/or helpful to the presentdescription.

Similarly, some drawings are simplified block diagrams showing onlypertinent components, and some of these components merely represent afunction and/or operation well-known in the field, rather than an actualpiece of hardware, as would be understood by one of ordinary skill inthe art. In such cases, some or all of the components/modules may beimplemented or provided in a variety and/or combinations of manners,such as at least partially in firmware and/or hardware, including, butnot limited to one or more application-specific integrated circuits(“ASICs”), standard integrated circuits, controllers executingappropriate instructions, and including microcontrollers and/or embeddedcontrollers, field-programmable gate arrays (“FPGAs”), complexprogrammable logic devices (“CPLDs”), and the like. Some or all of thesystem components and/or data structures may also be stored as contents(e.g., as executable or other machine-readable software instructions orstructured data) on a non-transitory computer-readable medium (e.g., asa hard disk; a memory; a computer network or cellular wireless networkor other data transmission medium; or a portable media article to beread by an appropriate drive or via an appropriate connection, such as aDVD or flash memory device) so as to enable or configure thecomputer-readable medium and/or one or more associated computing systemsor devices to execute or otherwise use or provide the contents toperform at least some of the described techniques.

One or more processors, simple microcontrollers, controllers, and thelike, whether alone or in a multi-processing arrangement, may beemployed to execute sequences of instructions stored on non-transitorycomputer-readable media to implement embodiments of the presentdisclosure. In some embodiments, hard-wired circuitry may be used inplace of or in combination with software instructions. Thus, embodimentsof the present disclosure are not limited to any specific combination ofhardware circuitry, firmware, and/or software.

The term “computer-readable medium” as used herein refers to any mediumthat stores instructions which may be provided to a processor forexecution. Such a medium may take many forms, including but not limitedto, non-volatile and volatile media. Common forms of non-transitorycomputer-readable media include, for example, a floppy disk, a flexibledisk, hard disk, magnetic tape, or any other magnetic medium, a CD-ROM,any other optical medium, punch cards, paper tape, any other physicalmedium with patterns of holes, a RAM, a PROM, and EPROM, a FLASH-EPROM,any other memory chip or cartridge, or any other medium on whichinstructions which can be executed by a processor are stored.

Some embodiments of the present disclosure may be implemented, at leastin part, on a portable device. “Portable device” and/or “mobile device”as used herein refers to any portable or movable electronic devicehaving the capability of receiving wireless signals, including, but notlimited to, multimedia players, communication devices, computingdevices, navigating devices, etc. Thus, mobile devices include (but arenot limited to) user equipment (UE), laptops, tablet computers, portabledigital assistants (PDAs), mp3 players, handheld PCs, instant messagingdevices (IMD), cellular telephones, global navigational satellite system(GNSS) receivers, watches, or any such device which can be worn and/orcarried on one's person.

Various embodiments of the present disclosure may be implemented in anintegrated circuit (IC), also called a microchip, silicon chip, computerchip, or just “a chip,” as would be understood by one of ordinary skillin the art, in view of the present disclosure. Such an IC may be, forexample, a broadband and/or baseband modem chip.

While several embodiments have been described, it will be understoodthat various modifications can be made without departing from the scopeof the present disclosure. Thus, it will be apparent to those ofordinary skill in the art that the present disclosure is not limited toany of the embodiments described herein, but rather has a coveragedefined only by the appended claims and their equivalents.

APPENDIX: EXPERIMENTAL VALIDATION A. Cascade Training

TABLE A-II Comparison of cascade training versus conventional trainingin Set14, scale3 PSNR SSIM CT-SRCNN 5-layer 29.44 0.8232 Non-CT-SRCNN5-layer 29.56 0.8258 CT-SRCNN 7-layer 29.50 0.8245 Non-CT-SRCNN 7-layer29.71 0.8287 CT-SRCNN 9-layer 29.52 0.8250 Non-CT-SRCNN 9-layer 29.750.8299 CT-SRCNN 13-layer 29.56 0.8265 Non-CT-SRCNN 13-layer 29.91 0.8324

In Table A-II, the PSNR/SSIM of a cascade trained CNN in accordance withthe present disclosure is compared to non-cascade trained CNN withunsupervised weight initialization from VDSR. It can be seen that withthe same network architecture, the PSNR/SSIM of CT-SRCNN is clearlybetter than non-cascade training.

FIG. 11 is an exemplary diagram illustrating the convergence speed ofcascade trained CNNs according to an embodiment of the presentdisclosure vs. non-cascade trained CNNs. The CT-SRCNN is found toconverge faster compared to non-CT-SRCNN. The accuracy of the CT-SRCNNconsistently increases when more layers are utilized. This indicatesthat cascade network training also trains SRCNNs deeper and deeper.Cascade network training performs better compared to conventionaltraining in both accuracy and convergence speed.

In Table A-III, the number of parameters, PSNR, SSIM, and time per imageof a CT-SRCNN-13 in accordance with the present disclosure is comparedto known SR networks in scale 3.

TABLE A-III Comparison of cascade training versus existing networks inSet14, scale3 Time per Number of Set 14 Set 14 image Parameters PSNRSSIM (in seconds) VDSR   >600,000 29.77 0.8314 0.17 DRCN >1,000,00029.76 0.8311 4.19 13-layer CT-SRCNN-13   ~150,000 29.91 0.8324 0.03

B. Cascade Network Trimming

Table A-IV shows that the cascade trimmed CT-SRCNN (where 4 out of the13 layers are trimmed) achieves similar performance to the non-cascadetrimmed CT-SRCNN, but the network size is reduced 20%. Cascade networktrimming according to the present disclosure is also applied to anothernetwork, namely, the FSRCNN (see Dong et al. 2016b). This networkconsists of 7 convolutional layers and one deconvolution layer. Similarto trimming the CT-SRCNN according to an embodiment of the presentdisclosure above, 2 layers of the FSRCNN are also trimmed in each stage.Table A-IV shows that network cascade trimming according to the presentdisclosure is also effective for FSRCNN.

TABLE A-IV Evaluation of cascade trimmed networks in Set14, scale3 Timeper Number of image Parameters PSNR SSIM (in seconds) CT-SRCNN 13 layer,~150,000 29.91 0.8324 0.03 no trimming Cascade trimmed 13-layer ~120,00029.91 0.8322 0.02 CT-SRCNN, trim 4 layers FSRCNN 8 layer,  ~12,000 29.520.8246 0.009 no trimming Cascade trimmed FSRCNN 8  ~8,500 29.51 0.82440.008 layer, trim 2 layers Cascade trimmed FSRCNN 8  ~6,800 29.35 0.82280.007 layer, trim 4 layers Cascade trimmed FSRCNN 8  ~4,900 29.35 0.82080.006 layer, trim 6 layers Cascade trimmed FSRCNN 8  ~3,400 29.22 0.81890.005 layer, trim 8 layers FSRCNN official lite version  ~3,900 29.170.8175 0.006

There is a trade-off between the trimming rate and the accuracy. If only2 layers (the 7^(th) and 8^(th)) are trimmed, there is almost noaccuracy loss, while 30% of the parameters are removed. If all 8 layersare trimmed (Cascade trimmed FSRCNN 8 layer, trim 8 layers), theaccuracy is still better compared to the official model (FSRCNN officiallite version), with a smaller network size (3,400 compared to 3,900parameters).

C. Dilated Convolution

Table A-V shows the experimental results of a dilated 13-layer CT-SRCNN.The dilation is applied for the first 9×9 layer, the second 5×5 layer,and the last 5×5 layer. Instead, 5×5, 3×3, and 3×3 2-dilatedconvolutional layers are utilized. It can be seen that the dilatedversion of CT-SRCNN can achieve similar PSNR/SSIM to the non-dilatedversion, but the network size is clearly reduced.

TABLE A-V Evaluation of dilated CT-SRCNN on Set14, scale 3 Number ofTime per image Parameters PSNR SSIM (in seconds) CT-SRCNN 13 layer~150,000 29.91 0.8324 0.03 Dilated CT-SRCNN 13 ~110,000 29.90 0.83240.02 layer

What is claimed is:
 1. An apparatus for generating a convolutionalneural network (CNN), the apparatus comprising: one or morenon-transitory computer-readable media; and at least one processorwhich, when executing instructions stored on the one or morenon-transitory computer-readable media, performs the steps of: startingcascade training on the CNN; and inserting one or more layers into theCNN where a training error converges or remains higher than a threshold.2. The apparatus of claim 1, wherein the at least one processor, whenexecuting instructions stored on the one or more non-transitorycomputer-readable media, performs the steps of: if the training errordoes not converge, determining whether the training error is less thanthe threshold; if the training error is less than the threshold,outputting the cascade trained CNN; and if the training error is notless than the threshold, starting a new stage.
 3. The apparatus of claim2, wherein a predetermined setting of each new layer in the new stage israndomly initialized with a Gaussian distribution with zero mean andstandard deviation σ.
 4. The apparatus of claim 1, wherein the CNN is asuper-resolution CNN (SRCNN) for processing at least one of images orvideo.
 5. The apparatus of claim 1, wherein the at least one processor,when executing instructions stored on the one or more non-transitorycomputer-readable media, performs the steps of: performing cascadetraining; and performing cascade network trimming after performingcascade training.
 6. The apparatus of claim 5, wherein cascade networktrimming comprises an iterative process of one or more stages, in whicheach stage comprises: trimming a set number of layers of the CNN byreducing dimensions of filters at one or more intermediate layers; ifthe training error is converging, determining whether all of the layersof the CNN have been trimmed; if all of the layers of the CNN have beentrimmed, outputting the network trimmed CNN; and if all of the layers ofthe CNN have not been trimmed, starting a new stage.
 7. The apparatus ofclaim 6, wherein cascade network trimming further comprises: if thetraining error is not converging, outputting the CNN at a cascadetrimming stage where the training error was last converging.
 8. Theapparatus of claim 1, wherein the cascade training is performed usingdilated convolutional filters.
 9. An apparatus for generating aconvolutional neural network (CNN), the apparatus comprising: one ormore non-transitory computer-readable media; and at least one processorwhich, when executing instructions stored on the one or morenon-transitory computer-readable media, performs the steps of:performing cascade network trimming of the CNN, the trimming comprising:trimming a number of layers of the CNN; and determining whether one ormore layers of the CNN have been trimmed.
 10. The apparatus of claim 9,wherein the cascade network trimming further comprises: determiningwhether a training error is converging; and if the training error is notconverging, outputting the CNN at a cascade trimming stage where thetraining error was last converging.
 11. The apparatus of claim 10,wherein each stage of the cascade network trimming further comprises:fine tuning before determining whether the training error is converging.12. The apparatus of claim 9, wherein trimming the number of layers ofthe CNN comprises, for each layer: trimming filters that do not meet acertain criteria.
 13. The apparatus of claim 12, wherein the certaincriteria comprises a measure of relative importance.
 14. The apparatusof claim 9, wherein the CNN is a super-resolution CNN (SRCNN) forprocessing at least one of images or video.
 15. The apparatus of claim9, wherein the at least one processor, when executing instructionsstored on the one or more non-transitory computer-readable media,performs the step of: performing cascade training of the CNN beforecascade network trimming.
 16. The apparatus of claim 15, wherein cascadetraining comprises an iterative process of one or more stages, in whicheach stage comprises: training the CNN; determining whether a trainingerror is converging; and if the training error is converging, insertinga preset number of intermediate layers in the CNN, wherein weights ofeach new layer are set to a predetermined setting; and starting a newstage.
 17. The apparatus of claim 16, wherein each stage of the cascadetraining iterative process further comprises: if the training error isnot converging, determining whether the training error is less than thethreshold; if the training error is less than the threshold, outputtingthe cascade trained CNN; and if the training error is not less than thethreshold, starting a new stage.
 18. The apparatus of claim 16, whereinthe predetermined setting of each new layer is randomly initialized witha Gaussian distribution with zero mean and standard deviation a.
 19. Theapparatus of claim 16, wherein the cascade training is performed usingdilated convolutional filters.
 20. An apparatus for generating aconvolutional neural network (CNN), the apparatus comprising: one ormore non-transitory computer-readable media; and at least one processorwhich, when executing instructions stored on the one or morenon-transitory computer-readable media, performs the steps of:performing cascade training on the CNN to produce a CNN output; andperforming network trimming of the CNN output.
 21. The apparatus ofclaim 20, wherein cascade training comprises an iterative process of oneor more stages, in which each stage comprises: training the CNN;determining whether a training error is converging; and if the trainingerror is converging, inserting a preset number of intermediate layers inthe CNN, wherein weights of each new layer being set to a predeterminedsetting; and starting a new stage.
 22. The apparatus of claim 20,wherein the cascade training is performed using dilated convolutionalfilters.
 23. The apparatus of claim 20, wherein the network trimmingcomprises an iterative process of one or more stages, in which eachstage comprises: trimming a set number of layers of the CNN by reducingdimensions of filters at one or more intermediate layers; determiningwhether a training error is converging; and if the training error isconverging, determining whether all of the layers of the CNN have beentrimmed; if all of the layers of the CNN have been trimmed, outputtingthe network trimmed CNN; and if all of the layers of the CNN have notbeen trimmed, starting a new stage.
 24. The apparatus of claim 20,wherein the CNN is a super-resolution CNN (SRCNN) for processing atleast one of images or video.
 25. A method of manufacturing a chipset,the method comprising: at least one processor which, when executinginstructions stored on one or more non-transitory computer-readablemedia, performs the steps of: performing cascade training on a CNN toproduce a CNN output; and performing network trimming of the CNN output.26. A method of testing an apparatus, the method comprising: testingwhether the apparatus has at least one processor which, when executinginstructions stored on one or more non-transitory computer-readablemedia, performs the steps of: performing cascade training on a CNN toproduce a CNN output; and performing network trimming of the CNN output.